Asymmetric Laplace laws and modeling financial data
Let \(v_{p}\) be a geometrically distributed random variable with mean \(1/p,\;p\in (0,1)\). A random vector \(Y\) has an asymmetric Laplace (AL) distribution in \(\mathbb R^{d}\) if and only if geometric compounds \(a_{p}\sum_{j=1}^{v_{p}}(X^{j}+b_{p})\), where i.i.d. random vectors \(X^{i}\) have finite variance, converge in distribution to \(Y\) as \(p\to 0\). If \(X^{i}\) have mean vector \(m\) and covariance matrix \(\Sigma\), then the limiting random vector \(Y\) has the characteristic function \(\Psi(t)=\left[1+{1\over 2}t'\Sigma t-it'm \right]^{-1}\). We use the notation AL\(_{d}(\Sigma,m)\) for the distribution given by the previous characteristic function. Let \(Y\sim\text{AL}_{d}(\Sigma,m)\) and \(X\sim N_{d}(0,\Sigma)\). Let \(Z\) be an exponentially distributed random variable with mean 1, independent of \(X\). Then the following representation holds: \(Y=mZ+Z^{1/2}X\). The main properties of AL laws are presented without proofs. The authors apply the AL model to the currency exchange rates data, showing good fit of this model. AL model is clearly better than normal distribution.
scientific article; zbMATH DE number 1755425
scientific article